validation & verification and uncertainty quantification ... system qualification ... economics,...

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,

for the United States Department of Energy’s National Nuclear Security Administration

under contract DE-AC04-94AL85000.

Validation & Verification and Uncertainty Quantification at Sandia

Brian M. Adams

Sandia National LaboratoriesOptimization and Uncertainty Quantification

July 18, 2008

Research Consortium for Multidisciplinary System Design Workshop

MIT, Boston, MA

Outline

•

Credible simulation and V&V•

Characterizing and propagating uncertainty for risk analysis and validation

•

Intro to aleatory

and epistemic UQ in DAKOTA•

Application examples:–

UQ for CMOS7 ViArray

UQ –

Sandia’s

QASPR program: computational model-

based system qualification

To be credible, simulations must be verified, validated with data, and deliver a best estimate of performance, together with its degree of variability or uncertainty.

Slide credits: Mike Eldred, Laura Swiler, Tony Giunta, Joe Castro, Genetha

Gray, Bill Oberkampf, Matt Kerschen, others

Insight from Computational Simulation

dHurricane Katrina: weather,

logistics, economics, human behavior

Electrical circuits: networks, PDEs, differential algebraic

equations (DAEs), E&M

Earth penetrator: nonlinear PDEs

with contact, transient analysis, material modeling

Micro-electro-mechanical systems (MEMS): quasi-

static nonlinear elasticity,

process modeling

Joint mechanics: system-

level FEA for component

assessment

Systems of systems analysis: multi-scale, multi-phenomenon

Credible Simulation

•

Ultimate purpose of modeling and simulation is (arguably) insight, prediction, and decision-making need credibility for intended application

•

Historically: primary focus on

modeling fidelity

Bill Oberkampf

Credible Simulation: V&V and UQ

Bill Oberkampf

Verification & Validation

•

Ultimately, quantification of margins and uncertainties (QMU): How close are my uncertainty-aware code predictions to required performance?

•

Validation:

“Are we solving the right equations?”–

A disciplinary science issue: is the science (physics, biology, etc.) model sufficient for the intended application?

–

Involves data and metrics; relies on uncertainty quantification (UQ)

•

Verification:

“Are we solving the equations correctly?”–

A mathematics/computer science issue:

is our mathematical formulation and software implementation of the physics model correct?

–

code verification

(software correctness): SQE, especially unit/regression/ verification testing; analytic problems, method of manufactured solutions

–

solution verification:

e.g., exhibits proper order of spatial/temporal/iterative convergence. Algorithms: Richardson extrapolation, finite element error estimation (ZZ, QOI)

Algorithms for Computational Modeling & Simulation

System Design

Geometric Modeling

Meshing

Physics

Model Equations

Discretization

Partitioning and Mapping

Nonlinear solve

Linear solve

Time integration

Information Analysis & Validation

AdaptOptimization

and UQ

Improved design and understanding

Are you sure you don’t need verification?!

Hierarchical Validation Experiments (Abnormal Thermal Environment)

SubassembliesSubassemblies

Full SystemFull System

SeparableEffectsSeparableEffects

ComponentsComponents

Deployed SystemDeployed System

Bill Oberkampf

increasing complexity, few

er experiments

Validation:

“Are we solving the right equations?”

Based on experimental data and metrics, is the model sufficient for the intended application?

Validation Metrics: Quantitative Comparison with Experiment

Bill Oberkampf

Validation Metrics: Quantitative Comparison with Experiment

Bill Oberkampf

F in a l T e m p e ra tu re V a lu e s

0

1

2

3

4

5

Te m e pra ture [de g C]

% in

Bin

Test Data

Model Data

Extrapolating Beyond Validation Domain

Uncertainties to Quantify

•

typical parametric uncertainty, incl. random fields/processes–

physics/science parameters–

statistical variation, inherent randomness–

operating environment, interference–

initial, boundary conditions; forcing–

geometry / structure / connectivity–

material properties–

manufacturing quality•

model form / accuracy•

program: requirements, technical readiness levels•

human reliability, subjective judgment, linguistic imprecision

•

numerical accuracy:

mesh, solver, approximation error•

experimental error:

measurement error, bias

A partial list of uncertainties affecting computational model results

•

A single optimal design or nominal performance prediction is often insufficient for –

decision making / trade-off assessment

–

validation with experimental data ensembles

•

Need to make risk-informed decisions, based on an assessment of uncertainty

Why Uncertainty Quantification?

Verification & Validation: A Formal, Iterative Process

•

Validation

is “the process

of determining the degree to which a computer model is an accurate representation of the real world from the perspective of the intended model applications.”

•

Relies on comparing code calculations to results of physical experiments, with

the goal of developing and quantifying confidence in codes to predict a specified problem result

•

Credibility

assesses model and experiment relevance, quantification and capture of non-

deterministic components, and model adequacy

CodeVerification

CodeVerification

DPApplication

DPApplication

PlanningPlanning

ExperimentDesign, Execution

& Analysis

ExperimentDesign, Execution

& Analysis

MetricsMetrics

AssessmentAssessment

Prediction & CredibilityPrediction

& Credibility

DocumentDocument

CalculationVerificationCalculationVerification

1

7

6

5

2

4

3

3

8

Trucano et al; SAND Report 2002-0341

Requirements and planning

Validation Experiments

Verif

icat

ion

Validation Metrics

Are we solving the equations correctly?

Are we using the correct equations?

Are we using the converged solution?

Well-characterized result: BEST ESTIMATE +

UNCERTAINTY

Coverage Matrix Shows Code Features Exercised in Verification Tests

Sierra Capabilities (subset) SNL-Problem LANL-Problem

Matrix helps prioritize gaps, create new verification problems to fill most important, w.r.t. intended use.

Fills Gap

Categories of Uncertainty

•

Aleatory

(think probability density function)–

Inherent variability (e.g., in a population)–

Irreducible uncertainty –

can’t reduce it by further knowledge

•

Epistemic (think bounded intervals)–

Subjective uncertainty–

Related to what we don’t know–

Reducible: If you had more data or more information, you could make your uncertainty estimation more precise

•

In practice, people try to transform or translate uncertainties to the aleatory

type and perform sampling and/or parametric analysis

Often useful algorithmic distinctions, but not always a clear division

•

based on uncertain inputs, determine variance of outputs and probabilities of failure (reliability metrics)

•

identify parameter correlations/local sensitivities, robust optima

•

identify inputs whose variances contribute most to output variance (global sensitivity analysis)

•

quantify uncertainty when using calibrated model to predict

Uncertainty QuantificationForward propagation: quantify the effect that uncertain (nondeterministic) input variables have on model output

Potential Goals:

Input Variables u

(physics parameters, geometry, initial and boundary conditions)

Computational

Model

Variable Performance

Measures f(u)

(possibly given distributions)

Output Distributions

N samples

measure 1

measure 2

Model

Typical method: Monte Carlo Sampling

u1

u2

u3

Uncertainty Quantification Example

•

Device subject to heating

(experiment or computational simulation)

•

Uncertainty in composition/ environment (thermal conductivity, density, boundary), parameterized by u1

, …, uN•

Response temperature f(u)=T(u1

, …, uN

)

calculated by heat transfer code

Given distributions of u1

,…,uN

, UQ methods calculate statistical info on outputs:•

Probability distribution of temperatures•

Correlations (trends) and sensitivity of temperature•

Mean(T), StdDev(T), Probability(T

≥

Tcritical

)

Final Temperature Values

0

1

2

3

4

5

30 36 42 48 54 60 66 72 78 84

Temeprature [deg C]

% in

Bin

UQ: Sampling Methods

Given distributions

of u1

,…,uN

, UQ methods…


0

1

2

3

4

5

30 36 42 48 54 60 66 72 78 84

Temeprature [deg C]

% in

Bin

Output Distributions

N samples

measure 1

measure 2

Model…calculate statistical info on outputs T(u1

,…,uN

)u1

u2

u3

• Monte Carlo sampling• Quasi-Monte Carlo• Centroidal

Voroni

Tessalation

(CVT)• Latin Hypercube sampling

0.2 0.2 0.2 0.2 0.2

A B C D−∞ ∞

0

0.2

0.4

0.6

0.8

1

A B C D−∞ ∞

Latin Hypercube Sampling (LHS)

•

Specialized Monte Carlo (MC) sampling technique: workhorse method in DAKOTA / at Sandia

•

Stratified random sampling among equal probability bins

for all 1-D projections of an n-dimensional set of samples.

•

McKay and Conover (early), restricted pairing by Iman

A B C D

G

H

I

J

K

L−∞ ∞

Intervals Used with a LHS of Size n = 5 in Terms of the pdf

and CDF for a Normal Random Variable

A Two-Dimensional Representation of One Possible LHS of size 5 Utilizing X1 (normal)

and X2 (uniform)


0

1

2

3

4

5

30 36 42 48 54 60 66 72 78 84

Temeprature [deg C]

% in

Bin

Calculating Probability of Failure

•

Given uncertainty in materials, geometry, and environment, determine likelihood of failure Probability(T

≥

Tcritical

)

•

Could perform 10,000 Monte Carlo samples and count how many exceed the threshold…

•

Or directly determine input variables which give rise to failure behaviors by solving an optimization problem.

Alternatives to Sampling

•

for a modest number of random variables, polynomial chaos expansions

may converge considerably faster to statistics of interest

•

if principal concern is with failure modes (tail probabilities),

consider global reliability methods

LHS sampling is robust, trusted, ubiquitous,

but advanced methods may offer advantages:

Hybrid

Surrogate Based

OptUnderUnc

Branch&Bound/PICO

Strategy

Optimization Uncertainty

2nd Order ProbabilityUncOfOptima

Pareto/Multi-Start

Upcoming (Mike): DAKOTA enables more efficient UQ by combining optimization, uncertainty analysis methods, and surrogate (meta-) modeling in a single framework.

Challenge: Epistemic UQ

•

Epistemic uncertainty:

insufficient information to specify a probability distribution

•

Subjective, reducible, or lack-of-knowledge uncertainty (given more resources to gather information, could reduce the uncertainty)

•

For example:–

“I expect this parameter to have a lognormal distribution, but only know bounds on its mean and standard deviation,”

or–

Dempster-Shafer belief structures: “basic probability assignment”

for each interval where the uncertain variable may exist; contiguous, overlapping, or gapped

BPA=0.5 BPA=0.2BPA=0.3 Variable 1

BPA=0.5 BPA=0.2BPA=0.3Variable 2

Propagating Epistemic UQ

Second-order probability–

Two levels: distributions/intervals on distribution parameters

–

Outer level can be epistemic (e.g., interval)–

Inner level can be aleatory

(probability distrs)–

Strong regulatory history (NRC, WIPP).

Dempster-Shafer theory of evidence–

Basic probability assignment (interval-based)–

Solve opt. problems (currently sampling-based)

to compute belief/plausibility for output intervals

New

New

Circuit UQ Analysis

Use DAKOTA with Xyce

circuit simulator to perform pre-

fabrication uncertainty analysis of new CMOS7 ViArray

•

ViArray: generic ASIC implementation platform•

Target applications: guidance, satellite, command & control•

Assess voltage droop/spike during photocurrent event•

Consider effect of process variation in each ‘layer’

on supply voltages; representative distributions:

•

Truncated normals

used for METAL and VIA; truncated lognormals used for CONTACT and polyc.

ViArray: Benefits of UQ

•

One ensemble of UQ calculations used to determine most sensitive parameters and output ranges: determined that sensitivity depends on final chip configuration

•

Suspicious UQ results led to correcting simulation failures not observed at nominal parameters

•

Gave process engineers and circuit designers insight into possible circuit behaviors

•

Sensitivity could help guide data collection

•

Ongoing work: assess interaction of package parasitics

with on-chip parasitics, V&V for photocurrent generation models

•

Military requirement: certify to hostile environment

neutrons create damageEmitter (n-type)

Base (p-type)

Collector (n-type)

xx

xxx

xx

damage degrades gain

Neutron Radiation Exposure Degrades Electronics

•

•

SPR dismantled end of FY06 to improve security posture

•



Base (p-type)

Collector (n-type)

xx

xxx

xx



pass/fail

testing

quantified

uncertainty

•

•

SPR dismantled end of FY06 to improve security posture

•



Base (p-type)

Collector (n-type)

xx

xxx

xx


QASPR (Qualification Alternatives to Sandia Pulse Reactor) methodology will certify qualification via modeling &

simulation with quantified uncertainty


UQ

M&SEC

select experiments in alternate facilities

γ,n – 100 mslong pulse

ion – 100 μsshort pulse

QASPR: Science-Based Engineering Methodology For Qualification

Risk InformedDecisions

QualificationEvidence2 4 6 8 10

time

0.2

0.4

0.6

0.8

1Current

uncertainty quantification

R2381.5k

Q28

MMBT2907

Q6MMBT2222

V23Vdc

M3

MTB30P06V

R239

1K

0

0

RF_PWR3_TXPA

Q62

BFS17A

DA_BATTERY3

R207

1k

R2361k

RL3

10

0

V8TD = 70msTF = 3nsTR = 3nsV1 = 3V2 = 0

Q63

MMBT2907

R24210

R251499

D50

MMSZ5236BT1

0

R204

4.99K

V1-8Vdc

FET_BIAS

VDD

R241

200

R240100

RF_PWR3_ENB_N

R206

1K

C15uF

R226

4.99k

R237

1k

R46

1k

V7TD = 0TF = 1msTR = 50msV1 = 0V2 = 10.8

0

C401uf

0

R210

10K

high performance, multi-fidelity, predictive computational modeling

validation

V&V for QASPR Components

•

Developing formal V&V plans

•

Each computational code subject to code and solution verification

•

UQ used to validate device model response against data ensembles

•

Ultimately systems (circuit) V&V for qualification

Device Prototype: UQ Extrapolation to SPR

•

Calibrated to other facilities, CHARON fills SPR gap

•

Uncertainty & bias characterized by 2 degrees of freedom–

facility multiplier–

device multiplier

•

Uncertainty quantified with D.O.E + statistical approach

End UQ Methodology Goal: Best Estimate + Uncertainty Prediction for SPR

Facility Multiplier, F

Device Multiplier, M

μM-MS

= 1.0

μF-SPR

= 1.0μF-ACR

= 0. 88

μM-FA

= 1.07

Model DevelopmentFacility Bias

BE+U Prediction

σF-SPR

Device Bias

σM-FA

+2σ

peak damage

-2σ

mean

Model Validation: Blind Prediction

UQ algorithms have a critical role in system validation

Transient Device Damage Response•

Fairchild response data within SPR hidden

•

First prototype

of the QASPR methodology (and real validation of the QASPR system)

•

Prediction + Uncertainty (+/-2σ

device and facility uncertainty)

Model Validation: Blind Prediction

UQ algorithms have a critical role in system validation

+/- 1-2% vertical error on experimental measurement

Transient Device Damage Response•

Fairchild response data within SPR hidden

•

First prototype

of the QASPR methodology (and real validation of the QASPR system)

•

Prediction + Uncertainty (+/-2σ

device and facility uncertainty)

Electrical Modeling Complexity

•

simple devices:

1 parameter, typically physical and measurable

•

e.g., resistor @ 100Ω

+/-

1%•

resistors, capacitors, inductors, voltage sources

Circuit Board

Large Digital Circuit(e.g., ASIC)

Sub-circuit (analog)

Single Device

device: 1 to 100s of params

sub-circuit: 10s to 100s of devices

ASIC: 1000s to millions of devices

•

complex devices:

many parameters, some physical, others “extracted”

(calibrated)•

multiple modes of operation•

e.g., zener

diode: 30 parameters, 3 bias states; many transistor models (forward, reverse, breakdown modes)

simulation tim

e grows exponentially

(G. Gray, M. M-C)

complex device models + replicates in circuits

UQ: Mitigate Explosion of Factors!

L

H

N

•

Consider bounding parameter sets?

•

Exploit natural hierarchy or network structure?

•

Use surrogate/macro-models as glue between levels?

•

Need approaches curbing the curse of dimensionality

Summary

•

Formal V&V process helps certify credible simulation

•

Uncertainty quantification algorithms are essential in validation and calibration under uncertainty

•

Complex, large-scale simulations demand research in advanced efficient UQ methods

Thank you for your [email protected]

http://www.sandia.gov/~briadam

To be credible, simulations must be verified, validated with data, and deliver a best estimate of performance, together with its degree of variability or uncertainty.

validation & verification and uncertainty quantification ... system qualification ... economics,...

Documents